Network Creation Games with 2-Neighborhood Maximization

TL;DR

This paper introduces the 2-neighborhood maximization game (2-NMG), a network formation model in which selfish agents pay a per-edge cost $\alpha$ to maximize the size of their 2-hop neighborhood. The authors establish structural and algorithmic results for Nash equilibria (NE) and greedy equilibria (GE), including constant diameter bounds (NE: $3$, GE: $4$ in the worst cases) that are independent of $\alpha$ and $n$, as well as nontrivial price-of-anarchy consequences. They prove the existence of diverse stable networks, including scalable base-and-satellite constructions with outdegree $\Omega(\log(n/\alpha))$ and $\Omega(n\log(n/\alpha))$ total edges, and show that best-response computation is NP-hard while improving-response cycles prevent the finite-improvement-property. The paper provides lower bounds $\Omega(\log(n/\alpha))$ for NE PoA and $\Theta(n)$ for GE PoA when $1\le\alpha\le2$, along with matching upper bounds in relevant regimes, highlighting a gap in performance between NE and GE. Together, these results yield a nuanced view of decentralized network formation under a 2-hop centrality objective and point to rich future directions, including extensions to general $\beta$ in star-celebrity settings.

Abstract

Network creation games are well-established for investigating the decentralized formation of communication networks, like the Internet or social networks. In these games, selfish agents that correspond to network nodes strategically create costly edges to maximize their centrality in the formed network. We depart from this by focusing on the simpler objective of maximizing the 2-neighborhood. This seems natural for social networks, as an agent's connection benefit is typically provided by her neighbors and their neighbors but not by strangers further away. For this natural model, we study the existence, the structure and the quality both of Nash equilibria (NE) and greedy equilibria (GE). We give structural results on the existence of degree-2 paths and cycles, and we provide tight constant bounds on the diameter. In contrast to most previous network creation game research, our bounds on the diameter are independent of edge cost $α$ and the number of agents $n$. Also, bounding the diameter does not imply bounding the price of anarchy, which calls for other methods. Using them, we obtain non-trivial bounds on the price of anarchy, including a $Ω(\log(\frac{n}α))$ lower bound for NE, and a tight linear bound for GE for low $α$.

Figures (6)

• Figure 1: Illustration of NE for various $\alpha \leq 2$ with consecutive nodes of a degree of $2$. The detailed descriptions can be found in Observations \ref{['obs:zweierketten']} and \ref{['obs:NEs']}.
• Figure 2: Illustration of a NE with $k=4$ (see Theorem \ref{['thm:regular-NE']}). Out of all $\binom{9}{5}$ sets of satellites, three are drawn exemplarily.
• Figure 3: A network used to show that GE with $|E| \in \Theta(n^2)$ exist (see Theorem \ref{['thm:GE-diam-3-quadratic-edges']}).
• Figure 4: Illustration of the NE used in Lemma \ref{['lem-NE-diam3-1alpha3']}. Nodes with multiple borders represent multiple identical nodes.
• Figure 5: Improving response cycle for the 2-NMG for any edge price $\alpha$.

Theorems & Definitions (48)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• Lemma 3
• Corollary 4: Follows from Lemma \ref{['lem-no-two-leaves-at-different-nodes']}
• Theorem 5
• Lemma 6
• proof